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48 Chapter 2 DISCRETE-TIME SIGNALS AND SYSTEMS
where z k ,k =1,...,N are N roots (also called natural frequencies) ofthe
characteristic equation
N∑
a k z k =0
0
This characteristic equation is important in determining the stability of
systems. If the roots z k satisfy the condition
|z k | < 1, k=1,...,N (2.23)
then a causal system described by (2.22) is stable. The particular part
of the solution, y P (n), is determined from the right-hand side of (2.21).
In Chapter 4 we will discuss the analytical approach of solving difference
equations using the z-transform.
2.4.1 MATLAB IMPLEMENTATION
A function called filter is available to solve difference equations numerically,
given the input and the difference equation coefficients. In its
simplest form this function is invoked by
y = filter(b,a,x)
where
b = [b0, b1, ..., bM]; a = [a0, a1, ..., aN];
are the coefficient arrays from the equation given in (2.21), and x is the
input sequence array. The output y has the same length as input x. One
must ensure that the coefficient a0 not be zero.
To compute and plot impulse response, MATLAB provides the function
impz. When invoked by
h = impz(b,a,n);
it computes samples of the impulse response of the filter at the sample
indices given in n with numerator coefficients in b and denominator coefficients
in a. When no output arguments are given, the impz function
plots the response in the current figure window using the stem function.
We will illustrate the use of these functions in the following example.
□ EXAMPLE 2.11 Given the following difference equation
y(n) − y(n − 1)+0.9y(n − 2) = x(n);
∀n
a. Calculate and plot the impulse response h(n) atn = −20,...,100.
b. Calculate and plot the unit step response s(n) atn = −20,...,100.
c. Is the system specified by h(n) stable?
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